Generalized Logistic Equations in Covid-Related Epidemic Models

Abstract

The logistic equation on population growth was proposed by Verhulst (Corresp Math Phys 10:113-126, 1838) [22], with the aim to provide a possible correction to the unrealistic exponential growth forecast by T. Malthus, (J Johnson, London, 1872) [13]. Population modeling became of particular interest in the 20${\text {th}}$ century to biologists urged by limited means of sustenance and increasing human populations. Verhulst’s scheme was rediscovered by A. Lotka, (Elements of Mathematical Biology. Dover, New York, 1956) [12], as a simple model of a self-regulating population. Subsequently, the use of logistic dynamics spreads across a huge number of different frameworks, especially in diffusion phenomena. The logistic differential equation is a fundamental element in quantitative study of population dynamics, its use also extends to the field of epidemiology: both to describe the evolution of the infected population in deterministic models, and working in conditions of uncertainty it is the deterministic component of stochastic differential equations. This work brings a contribution to the foundational basic research on the logistic equation and its generalizations which hopefully have repercussions for epidemiologic applications.